Detection of rotational and involutional symmetries and congruity of polyhedra

We propose a simple and efficient general algorithm for determining both rotational and involutional symmetries of polyhedra. It requiresO(m ²) time and usesO(m) space, wherem is the number of edges of the polyhedron. As this is the lower bound of the symmetry detection problem for the considered output form, our algorithm is optimal. We show that a slight modification of our symmetry detection algorithm can be used to solve the related conguity problem of polyhedra.     

Analyzing skewed symmetries

Symmetry is pervasive in both man-made objects and nature. Since symmetries project to skew symmetries, finding axes of skew symmetry is an important vision task. This paper presents a linear time algorithm for finding the axes of skew symmetry, where the degree of symmetry is known. First, we present a review and critique of current methods for finding the axes of skew symmetry. Next, we decompose the problem of finding skew symmetry into the subproblems of solving for the rotational parameter of a shear symmetry and recovering the shear parameter of a reflexive symmetry. Using this approach, the authors derive a direct, non-heuristic moment-based technique for finding the axes of skew symmetry. For skew symmetric figures with degree of symmetry less than five we obtain a closed-form solution. The method does not rely on continuous contours but assumes there is no occlusion and requires knowing the contour's degree of symmetry. It is the first algorithm to find the axes of skew symmetry inO(n) time, where n is the number of contour points. The method is especially suited to industrial applications where the degree of symmetry is often knowna priori. Examples of the method are presented for both real and synthetic images, and an error analysis of the method is given.     

Conjugate symmetry

Conjugate symmetry is an entirely new approach to symmetric Boolean functions that can be used to extend existing methods for handling symmetric functions to a much wider class of functions. These are functions that currently appear to have no symmetries of any kind. Conjugate symmetries occur widely in practice. In fact, we show that even the simplest circuits exhibit conjugate symmetry. To demonstrate the effectiveness of conjugate symmetry we modify an existing simulation algorithm, the hyperlinear algorithm, to take advantage of conjugate symmetry. This algorithm can simulate symmetric functions faster than non-symmetric ones, but due to the rarity of symmetric functions, this optimization is of limited benefit. Because the standard benchmark circuits contain many symmetries it is possible to simulate these circuits faster than is possible with the fastest known event-driven algorithm. The detection and exploitation of conjugate symmetries makes use of GF(2) matrices. It is likely that conjugate symmetry and GF(2) matrices will find applications in many other areas of EDA.     

Symmetry identification of a 3-D object represented by octree

An algorithm for identifying symmetry of a 3-D object given by its octree is presented, and the symmetry degree (a measure of object symmetry) is proposed. The algorithm is based on traversals of the octree obtained by the principal axis transform of an input octree. An object can be in an arbitrary position and with arbitrary orientation within the octree space, and a wide range of symmetries represented by groups of proper and improper rotations can be identified. It is shown that the octree data structure supports these operations well, especially for objects whose symmetry types are simpler or equal in complexity with a fourfold rotational symmetry. The operation of the algorithm is illustrated using some synthetic test objects. The results, which are composed of identified symmetry types and the corresponding symmetry degrees, were satisfactory     

Static and dynamic structural symmetry breaking

We reconsider the idea of structural symmetry breaking for constraint satisfaction problems (CSPs). We show that the dynamic dominance checks used in symmetry breaking by dominance-detection search for CSPs with piecewise variable and value symmetries have a static counterpart: there exists a set of constraints that can be posted at the root node and that breaks all the compositions of these (unconditional) symmetries. The amount of these symmetry-breaking constraints is linear in the size of the problem, and yet they are able to remove a super-exponential number of symmetries on both values and variables. Moreover, we compare the search trees under static and dynamic structural symmetry breaking when using fixed variable and value orderings. These results are then generalised to wreath-symmetric CSPs with both variable and value symmetries. We show that there also exists a polynomial-time dominance-detection algorithm for this class of CSPs, as well as a linear-sized set of constraints that breaks these symmetries statically.     

How to calculate symmetries of Petri nets

Symmetric net structure yields symmetric net behaviour. Thus, knowing the symmetries of a net, redundant calculations can be skipped. We present a framework for the calculation of symmetries for several net classes including place/transition nets, timed nets, stochastic nets, self–modifying nets, nets with inhibitor arcs, and many others. Our approach allows the specification of different symmetry groups. Additionally it provides facilities either to calculate symmetries on demand while running the actual analysis algorithm, or to calculate them in advance. For the latter case we define and calculate a ground set of symmetries. Such a set has polynomial size and is sufficient for an efficient implementation of the for all symmetries loop and the partition of net elements into equivalence classes. These two constructions are the usual way to integrate symmetries into an analysis algorithm. Received 7 July 1997 / 10 August 1999     

Excluding Symmetries in Constraint-Based Search

We introduce a new method, called symmetry excluding search (SES), for excluding symmetries in constraint based search. To our knowledge, it is the first declarative method that can be applied to arbitrary symmetries. The SES-method is based on the notion of symmetric constraints, which are used in our modification of a general constraint based search algorithm. The method does not influence the search strategy. Furthermore, it can be used with either the full set of symmetries, or a subset of all symmetries.We proof correctness, completeness and symmetry exclusion properties of our method. Then, we show how to apply the SES-method in the special case of geometric symmetries (rotations and reflections) and permutation symmetries. Furthermore, we give results from practical applications.     

Interactive color image segmentation with linear programming

Image segmentation is an important and fundamental task for image and vision understanding. This paper describes a linear programming (LP) approach for segmenting a color image into multiple regions. Compared with the recently proposed semi-definite programming (SDP)-based approach, our approach has a simpler mathematical formulation, and a far lower computational complexity. In particular, to segment an image of M × N pixels into k classes, our method requires only O((M N k) ^m ) complexity—a sharp contrast to the complexity of O((M N k)²ⁿ ) if the SDP method is adopted, where m and n are the polynomial complexity of the corresponding LP solver and SDP solver, respectively (in general we have m≪ n). Such a significant reduction in computation readily enables our algorithm to process color images of reasonable sizes. For example, while the existing SDP relaxation algorithm is only able to segment a toy-size image of, e.g., 10 × 10 to 30 × 30 pixels in hours time, our algorithm can process larger color image of, say, 100 × 100 to 500 × 500 image in much shorter time.     

A simple and efficient algorithm for determining the symmetries of polyhedra

In this paper we present a simple and efficient algorithm for determining the rotational symmetries of polyhedral objects in O(m²) time using O(m) space, where m represents the number of edges of the object. Our algorithm is an extension of Weinberg's algorithm for determining isomorphisms of planar triply connected graphs. The symmetry information detected by our algorithm can be utilized for various purposes in artificial intelligence, robotics, assembly planning and machine vision.     

More Efficient Topological Sort Using Reconfigurable Optical Buses

Topological sort of an acyclic graph has many applications such as job scheduling and network analysis. Due to its importance, it has been tackled on many models. Dekel et al. [3], proposed an algorithm for solving the problem in O(log² N) time on the hypercube or shuffle-exchange networks with O(N ³) processors. Chaudhuri [2], gave an O(log N) algorithm using O(N ³) processors on a CRCW PRAM model. On the LARPBS (Linear Arrays with a Reconfigurable Pipelined Bus System) model, Li et al. [5] showed that the problem for a weighted directed graph with N vertices can be solved in O(log N) time by using N ³ processors. In this paper, a more efficient topological sort algorithm is proposed on the same LARPBS model. We show that the problem can be solved in O(log N) time by using N ³/log N processors. We show that the algorithm has better time and processor complexities than the best algorithm on the hypercube, and has the same time complexity but better processor complexity than the best algorithm on the CRCW PRAM model.     

Transformational Derivation of an Improved Alias Analysis Algorithm

In this paper we use a program transformational approach to obtain an asymptotically improved may-alias analysis algorithm. We derive an O(N³) time algorithm for computing an intra-procedural flow sensitive may-alias analysis, where N denotes the number of edges in the program control flow graph (CFG). Our algorithm improves the previous O(N⁵) time algorithm by Hind et al. [19]. Our time complexity improvement comes without any deterioration in space complexity. We also show that for a large subclass of programs in which the in-degree and out-degree of all CFG nodes is bounded by a constant, our algorithm is linear in the sum of the number of edges in the CFG of the program and the size of the output, i.e., the size of the computed alias information, and is therefore asymptotically optimal. Our transformational algorithm derivation technique also leads to a simplified yet precise analysis of time complexity.The work in this paper was done when the author was a graduate student at New York University. This paper was originally submitted when the author was a Research Staff Member at the IBM T.J. Watson Research Center.     

A faster algorithm for sorting on mesh-connected computers with multiple broadcasting using fewer processors

This paper presents an efficient parallel algorithm for sorting N data items on two-dimensional mesh connected-computers with multiple broadcasting (2-MCCMB). The algorithm uses N × N ^2/3 processors and takes 0(N ^1/3) time, whereas the previous algorithm by Chung-Horng Lung [3] uses N × N processors and takes 0(N ^l/2) time on 2-MCCMB.     

Efficient Algorithms for the Riemann-Roch Problem and for Addition in the Jacobian of a Curve

We study the effective Riemann-Roch problem of computing a basis for the linear space L(D) associated with a divisor D on a plane curve C and its applications. Our presentation begins with an extension of Noether's algorithm for this problem to plane curves with singularities. Like the original, this algorithm has a worst-case complexity of Ω(n!^|D|), where n is the degree of the curve C.We next consider representations of divisors based on Chow forms. Using these, we produce a factorization-free polynomial-time algorithm for the effective Riemann-Roch problem, which improves the complexity of Noether's algorithm by an order of magnitude. We also present further improvements which, for curves of bounded degree, yield an algorithm with complexity which is linear in the size of the given divisor. Finally, we consider applications of this problem: to the arithmetic of the Jacobian of a plane curve, to the construction of rational functions on a curve with prescribed zeroes and poles, and to the construction of curves with prescribed intersections.     

六边形稀疏网格上的FFT算法

稀疏网格是一种具有特殊分层插值性质的非均匀网格形式,稀疏网格上的离散傅立叶变换算法称为Hyperbolic Cross FFT算法.这一算法能够有效降低采样点数量,并将指数时间复杂度的d维DFT算法降低到O(Nlog^dN)^[10].六边形网格是另一种具有特殊性质的网格,具有在采样点数量较少和采样效率较高等优势.本文的研究工作主要集中在将六边形网格和稀疏网格相结合,构造六边形稀疏网格上的FFT算法.通过定义六边形和方形网格下标之间的转换,实现了六边形稀疏网格上的FFT算法,并通过数值实验证明了这一算法的有效性.     

并行马尔可夫随机场计算变分光流方法

目的 基于马尔可夫随机场（MRF）的变分光流计算是一种较为鲁棒的光流计算方法,但是计算效率很低。置信传播算法（BP） 是一种针对MRF较为高效的全局优化算法。本文提出一种MRF变分光流计算模型并采用并行BP方法实现,极大提高计算效率。方法 提出的MRF变分光流计算模型中的数据项采用了Horn等人根据灰度守恒假设得到的光流基本约束方程,并采用非平方惩罚函数进行调整以平滑边界影响。为在CUDA平台上实现高效并行处理,本文提出了一种优化的基于置信传播的MRF并行光流计算方法。该优化方法在采用置信传播最小化MRF光流能量函数时,采用了一种4层的3维网络结构进行并行计算,每层对应MRF4邻域模型中的一个方向的信息传播,同时在每层中为每个像素分配多个线程采用并行降维法计算所要传递的信息,大大降低单线程计算负荷,大幅度提高计算效率。结果 采用旋转小球图像序列进行实验,计算效率提高314倍;采用旋转小球、Yosemite山谷和RubberWhale 3种不同图像序列,与Horn算法、Weickert算法、Hossen并行Lucas算法、Grauer-Gray并行MRF算法进行对比实验,本文方法得到最低的平均端点误差（AEE）,分别为0.13、0.55和0.34。结论 本文提出了一种新的MRF光流计算模型,并在CUDA平台上实现了并行优化计算。实验结果表明,本文提出的并行计算方法在保持计算精度的同时极大提高了计算效率。本文方法对内存需求巨大,在处理高分辨率图像时,限制了采样点数,难以计算大位移。     

Optimal algorithms for finding the symmetries of a planar point set

We present an asymptotically optimal algorithm to locate all the axes of mirror symmetry of a planar point set. The algorithm was derived by reducing the 2-D symmetry problem to linear pattern-matching. Optimal algorithms for finding rotational symmetries and deciding whether a point symmetry exists are also presented.     

Finding rotational symmetries by cyclic string matching

Symmetry is an important shape feature. In this paper, a simple and fast method to detect perfect and distorted rotational symmetries of 2D objects is described. The boundary of a shape is polygonally approximated and represented as a string. Rotational symmetries are found by cyclic string matching between two identical copies of the shape string. The set of minimum cost edit sequences that transform the shape string to a cyclically shifted version of itself define the rotational symmetry and its order. Finally, a modification of the algorithm is proposed to detect reflectional symmetries. Some experimental results are presented to show the reliability of the proposed algorithm.     

A network model of barrier synchronization algorithms

We consider two algorithms for the barrier synchronization ofN processes: the Dissemination algorithm⁽²⁾ and Brooks algorithm.^(1,2) Both algorithms comprise a number of binary communications amongst the processes, organized into a sequence of stages. It is shown that Brooks' algorithm⁽¹⁾ requires between LogN(log₂ N) and 2 LogN stages, the lower bound being guaranteed only in the case thatN is a power of 2 (cubic) and the upper bound seemingly needed for most otherN. On the other hand, it is shown⁽²⁾ that the Dissemination algorithm requires only LogN stages regardless ofN, making it apparently superior to the Brooks algorithm. We introduce a network model of local barrier algorithms. Using it we obtain a rigorous correctness proof for local barrier algorithms, and show that the number of stages in the Brooks algorithm is bounded above by LogN+1. The Brooks algorithms is therefore essentially equivalent in time complexity to the Dissemination algorithm. We then address the question of which values ofN admit exactly LogN Brooks stages. We find a sufficient condition, and conjecture that it is also necessary.